Residual and Past Entropy in Actuarial Science and Survival Models

Methodol Comput Appl Probab (2014) 16:79–99 DOI 10.1007/s11009-012-9300-0

Residual and Past Entropy in Actuarial Science and Survival Models Athanasios Sachlas · Takis Papaioannou

Received: 9 November 2011 / Revised: 3 July 2012 / Accepted: 19 July 2012 / Published online: 15 August 2012 © Springer Science+Business Media, LLC 2012

Abstract The best policy for an insurance company is that which lasts for a long period of time and is less uncertain with reference to its claims. In information theory, entropy is a measure of the uncertainty associated with a random variable. It is a descriptive quantity as it belongs to the class of measures of variability, such as the variance and the standard deviation. The purpose of this paper is to investigate the effect of inflation, truncation or censoring from below (use of a deductible) and truncation or censoring from above (use of a policy limit) on the entropy of losses of insurance policies. Losses are differentiated between per-payment and perloss (franchise deductible). In this context we study the properties of the resulting entropies such as the residual loss entropy and the past loss entropy which are the result of use of a deductible and a policy limit, respectively. Interesting relationships between these entropies are presented. The combined effect of a deductible and a policy limit is also studied. We also investigate residual and past entropies for survival models. Finally, an application is presented involving the well-known Danish data set on fire losses. Keywords Entropy · Loss distributions · Truncation and censoring · Residual and past entropy · Proportional hazards · Proportional reversed hazards · Frailty models AMS 2000 Subject Classifications 62B10 · 62P05

A. Sachlas (B) · T. Papaioannou Department of Statistics & Insurance Science, University of Piraeus, 80 Karaoli & Dimitriou Street, 185 34, Piraeus, Greece e-mail: [email protected] T. Papaioannou e-mail: [email protected]

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1 Introduction In most insurance policies the amount of money paid by the insurer following a claim is not necessarily all but it is part of the loss that occurs. There are several standard types of partial insurance coverage such as insurance with a deductible, insurance with a policy limit, residual insurance etc. The best policy for an insurance company is that which lasts for a long period of time and is less uncertain with reference to its claims. A well known measure of uncertainty associated with a random variable comes from the field of information theory and is called entropy. The most widely known measure of entropy, Shannon’s entropy, is given by   H(X) = − p(x) ln p(x) or H(X) = − f (x) ln f (x) dx x

depending on whether the random variable X is discrete or continuous, respectively, where p(x) is the probability mass function and f (x) the probability density function (pdf) of X (Shannon 1948). In the latter case H(X) is also called differential entropy. In the sequel H(X) will also be denoted by H X . In this paper we shall restrict to non-negative random variables. If the random variable X is censored from below at point d, i.e. we observe Y(d) = max{X, d}, the entropy of Y(d), which is a mixed variable consisting of a discrete part at d and a continuous part over (d, +∞), will be defined by  ∞ H(Y(d)) = − f (x) ln f (x) dx − F(d) ln F(d), (1) d

where F(x), x > 0 is the cumulative distribution function (cdf) of X. It is the sum of the continuous and discrete entropies since at d we have placed a probability mass of size F(d). This definition is consistent with our intuition as well as with similar definitions given in Baxter (1989) and Nair et al. (2006). If X is censored from above or censored both from below and above the entropy is defined analogously. The ¯ p + p¯ = 1 will be denoted by entropy of a Bernoulli distribution ( p, p), ¯ = − p ln p − p¯ ln p. ¯ H[ p, p]

(2)

Entropy quantifies the expected uncertainty related with the result of an experiment or in other words it provides information about the predictability of the result of a random variable X. The larger the entropy the less concentrated the distribution of X and thus an observation on X provides little information. Shannon’s entropy, along with other measures of entropy such as the Rényi entropy, may be regarded as a descriptive quantity of the corresponding pdf. The entropy serves as a measure of variability for continuous variables or as a measure of variation or diversity of the possible values of a discrete variable (Harris 1982; Nadarajah and Zografos 2003; Zografos 2008). There exists a large literature on entropy, its axiomatic foundation and its properties, applications and generalizations. Among them prominent place occupies the maximum entropy principle which provides characterizations of distributions and leads to probability models. Other measures of entropy, and generalizations thereof, have been proposed as well (Cover and Thomas 1991; Pardo 2006). As such, derivations of explicit expressions for the Shannon and

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other entropies for univariate and multivariate distributions has been a subject of interest. Highly uncertain insurances are less reliable. The uncertainty for the loss in an insurance policy can be quantified by the entropy of the corresponding loss distribution. However, frequently in actuarial practice, the practitioner has at hand transformed data as a consequence of deductibles and policy (liability) limits. The primary purpose of this paper is to investigate the uncertainty under several partial insurance schemes. In Section 2 we investigate the effect of inflation on entropy, in Section 3 we investigate the effect of a deductible while the effect of policy limits is investigated in Section 4. In Section 5 we investigate the combined effect of a deductible and a policy limit. The entropy of several survival models such as the proportional hazards, the proportional reversed hazards and the frailty model is presented in Section 6. In Section 7 we provide an application, numerical illustration of the entropic results using the well-known data set of losses in Danish fires. Finally concluding remarks are given in the last section. In our examples, we shall use the following loss distributions: Exponential with τ f (x) = λe−λx , λ > 0, x > 0, Weibull with f (x) = cτ xτ −1 e−cx , c > 0, τ > 0, x > 0, and Pareto with f (x) = αλα (λ + x)−(α+1) , α > 0, λ > 0, x > 0. The incomplete gamma ∞ −t function  ∞ −t (0, x) = x e /t dt = E1 (x) and the exponential integral function Ei(x) = − −x e /t dt (Abramowitz and Stegun 1972, pp. 262–263) appear in some of the entropy expressions. For other loss distributions we refer to Kluggman et al. (2008).

2 The Effect of Inflation The basic effect of inflation is an increase in losses. An existing financial and actuarial model must often be adjusted to bring it up to the current level of loss experience since, in general, the model was estimated from observations made years in the past. In addition, we may desire a projection to reflect losses anticipated in some future period. In the sequel we will investigate the effect of inflation on differential entropy. Let X be a random variable that models losses a certain year. The random variable that models losses after one year and under the inflation effect is X(r) = (1 + r)X, where r, r > 0, is the annual inflation rate. In other words if X is the value of losses and r the annual inflation rate, one year after the loss will have a value of X(r) = (1 + r)X. The pdf of X(r) is given by   1 z f X(r) (z) = fX , z ∈ R, 1+r 1+r where f X is the pdf of X. The relation between the entropies of X and X(r) = (1 + r)X, is given in the following lemma. Lemma 1 The entropy of losses X(r) = (1 + r)X after one year and under inf lation with annual inf lation rate r, r > 0, is given by H X(r) = H X + ln(1 + r), where H X is the entropy of X.

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Proof The above lemma is an application of Theorem 9.6.4 of Cover and Thomas (1991, p. 233) and its proof is omitted.   The extension to k years is obvious. Explicit expressions for Shannon’s entropy under inflation and for various loss distributions can be obtained by just adding the constant ln(1 + r) to the loss distributions’ entropy expressions. A useful proposition is following: Proposition 1 For r > 0, the differential entropy of X(r) is always larger than that of X and is an increasing function of r. The previous proposition says that, as expected, in the case of loss models under inflation, we will always have more uncertainty compared with the losses without inflation. Furthermore, the uncertainty for losses increases when the inflation rate increases. Specifically, since differential entropy is considered as a descriptive measure of the variability of a distribution, the more the inflation increases the more the variability of losses or the risk of returns increases.

3 The Effect of a Deductible Suppose that losses are not recorded or reported below a specified amount. In this case the data are referred to as truncated from below or left truncated. The most common reason of left truncation is the use of a deductible. Let X be the absolutely continuous non-negative random variable of losses with pdf f X (x), cdf F X (x), x > 0, survival function F¯ X (x) = 1 − F X (x), and d > 0 the deductible value. Following the terminology of Kluggman et al. (2008), the random variable expressing losses associated with an insurance policy having a deductible d is given in two ways: (i) per-payment (what is actually paid by the policy; there is a loss but no benefits are paid to the policyholder if loss is less than d)  X, X>d Y p (d) = equivalently Y p (d) = X|X > d not defined, X ≤ d (ii) per-loss (franchise deductible, actual amount paid by the insurance company)  X, X > d Yl (d) = 0, X ≤ d. In the first case we have truncation from below while in the second we have censoring from below. The difference between these two random variables is that the latter assigns a positive probability mass at the zero point (when X ≤ d) that makes it not absolutely continuous. It becomes a mixed variable consisting of a discrete and a continuous part. In the per-payment case, losses or claims less than the deductible may not be reported to the insurance company. In the per-loss case, all losses or claims are reported. The per-payment variable is the per-loss one given that the latter is positive. The per-loss case expresses real losses for the insurer. Let us now investigate the effects of truncation and censoring from below. Differential entropy in this situation is a useful measure as it measures the uncertainty about the loss covered by the insurance company. Lemma 2 gives the relation

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between the entropies of X and Y p (d) while Lemma 3 gives the relation between the entropies of X and Yl (d). Lemma 2 The per-payment entropy of losses with a deductible d, denoted by H( f, d), is given by  ∞ f X (x) f X (x) ln dx H( f, d) = − ¯ ¯ F X (d) F X (d) d ⎡ ⎤ d 1 ⎣ = HX + f X (x) ln f X (x) dx⎦ + ln F¯ X (d). F¯ X (d) 0

Proof It is known that the pdf of Y p (d) is given by ⎧ ⎨ f X (x) , x > d fY p (d) (x) = F¯ X (d) ⎩ 0, x ≤ d, Thus the entropy of Y(d) is given by   ∞  ∞ 1 ¯ H( f, d) = − f X (x) ln f X (x) dx − f X (x) ln F X (d) dx F¯ X (d) d d    d 1 ¯ ¯ = f X (x) ln f X (x) dx + F X (d) ln F X (d) HX + F¯ X (d) 0    d 1 = f X (x) ln f X (x) dx + ln F¯ X (d), HX + F¯ X (d) 0 where F¯ X (d) = 1 − F X (d) is the “survival function” at point d.

 

For brevity, H( f, d) will be called the per-payment entropy with a deductible d. Lemma 3 The per-loss entropy of losses with a deductible d, denoted by Hl ( f, d), is given by  ∞ Hl ( f, d) = − f X (x) ln f X (x) dx − F X (d) ln F X (d) d



= HX +

d

f X (x) ln f X (x) dx − F X (d) ln F X (d).

0

Proof The entropy of Yl (d) is obtained by adding the continuous and discrete part in accordance with definition in Eq. 1. Note that P(Yl (d) = 0) = F X (d). The rest is obvious.   For brevity, Hl ( f, d) will be called the per-loss entropy with a deductible d. Both H( f, d) and Hl ( f, d) can be negative as well as −∞ or +∞. Also they are not always less than H X as one may expect.

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If losses for an insurance policy exceed the deductible then H( f, d) measures the expected uncertainty of losses. Unconditionally, we are interested in the per-loss case. In the authors opinion the per-loss case is the most amenable to the analysis of uncertainty. We now give examples for the newly introduced entropies. Their expressions are derived by direct integration from their definitions using standard techniques. Where necessary, integration by parts or change of variables are employed. The same holds for the rest of examples of the paper. The original entropies are available in the literature. Example 1 The per-payment entropy with a deductible d for the exponential distribution is H( f, d) = 1 − ln λ, which is the same as the original entropy of the exponential distribution. Thus, use of a deductible does not affect the entropy of losses following the exponential distribution. The per-payment entropy with a deductible d for the Weibull distribution is τ

H( f, d) = 1 − ln c − ln τ − (τ − 1) ln d − (τ − 1)ecd (0, cdτ )/τ. (γ + If τ > 1 then H( f, d) < H X for all c and d, where H X = 1 − ln c − ln τ + τ −1 τ ln c). For d > 0, H( f, d) is increasing in d if τ < 1 and decreasing in d if τ > 1. The same entropy for the Pareto distribution is  −α 1 λ H( f, d) = (2 + 1/α + ln λ) + 1 + + ln(λ + d) − ln α. λ+d α Note that H( f, d) > H X , for all d and all α and λ > 1, where H X = 1 + ln α. Also H( f, d) is increasing in d for all d.

1 α

+ ln λ −

Example 2 The per-loss entropy with a deductible d for the exponential distribution is Hl ( f, d) = e−λd (1 + λd − ln λ) − (1 − e−λd ) ln(1 − e−λd ). In this case the per-loss entropy depends on d. The per-loss entropy with a deductible d for the Weibull distribution is τ

Hl ( f, d) = e−cd (1 − ln c − ln τ − (τ − 1) ln d + cdτ ) − (τ − 1)(0, cdτ )/τ   τ τ − 1 − e−cd ln 1 − e−cd . The same entropy for the Pareto distribution is  α λ Hl ( f, d) = (1 + 1/α − ln α − α ln λ + (α + 1) ln(λ + d)) λ+d   α   α   λ λ − 1− ln 1 − . λ+d λ+d The per-payment and per-loss entropies for other loss models such as Gamma, Transformed Gamma, Lognormal etc. are quite complicated as they involve higher order functions such as the Meijer function, the error function Erf(z), etc. (Abramowitz and Stegun 1972). For some models, such as the Burr distribution and the Generalized Pareto distribution the Shannon entropy has no closed form, so a numerical method should be employed in order to compute their value.

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From the expressions in Lemmas 2 and 3, it follows that the per-payment H( f, d) and per-loss Hl ( f, d) entropies are connected with the following relationship Hl ( f, d) = F¯ X (d)H( f, d) + H[F X (d), F¯ X (d)].

(3)

Also comparing the expressions in the previous lemmas termwise we obtain the following proposition. Proposition 2 If

∞ d

f X (x) ln f X (x)dx > 0, then H( f, d) < Hl ( f, d) for all d.

Proof It is not hard to see that  Hl ( f, d) − H( f, d) = H X + 0

−

1 F¯ X (d)

d

f X (x) ln f X (x)dx − F X (d) ln F(d) −



d

1 HX ¯ F X (d)

f X (x) ln f X (x)dx − ln F¯ X (d)

0

   d F X (d) =− f X (x) ln f X (x)dx HX + F¯ X (d) 0 −F X (d) ln F X (d) − ln F¯ X (d)    d F X (d) >− f X (x) ln f X (x)dx HX + F¯ X (d) 0  F X (d) ∞ = f X (x) ln f X (x)dx. F¯ X (d) d   Related to the previous losses with a deductible d are the residual (after d) losses defined by X − d (Kluggman et al. 2008): (i) per-payment residual loss  X − d, X>d equivalently Z p (d) = X − d|X > d Z p (d) = not defined, X ≤ d (ii) per-loss residual loss  X − d, X > d Z l (d) = equivalently Z l (d) = max{X − d, 0}. 0, X≤d Note that X − d is the retention loss (Cox 1991). The first case is truncation from below with shifting while the second is censoring from below with shifting. In these situations the insurer pays the amount of loss in excess of d if the loss is greater than d. In the second case the insurer pays zero for losses smaller than d. Again Z l (d) is not an absolutely continuous random variable. It is composed by an absolutely continuous and a discrete part.

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Lemma 4 (i) The per-payment residual entropy of losses with a deductible d is identical with the (simple) per-payment entropy. (ii) The per-loss residual entropy of losses with a deductible d is identical with the (simple) per-loss entropy. Proof Immediate since entropy is invariant in shift or location changes (Cover and Thomas 1991).   It is obvious that if losses X are larger than the deductible d, H( f, d) and Hl ( f, d) measure the expected uncertainty in the remaining or residual losses X − d, perpayment or per-loss respectively. The per-payment simple or residual entropy H( f, d) with a deductible d can be expressed in terms of the hazard or risk function λ X (x) = f X (x)/ F¯ X (x), x > 0 of X. Integrating by parts the integral defining H( f, d) we have  ∞ 1 H( f, d) = 1 − (4) (ln λ X (x)) f X (x) dx. F¯ X (d) d The above equation is identical to Eq. 1.2 of Ebrahimi and Pellerey (1995) in the framework of lifetime distributions. The connection with insurance losses will be given below. Obviously, Hl ( f, d) cannot be expressed in terms of hazard since Yl (d) is not absolutely continuous. Proposition 3 H( f, d) is independent of d if and only if λ X (x) is constant. Proof Assuming that λ X (d) = λ constant, Eq. 4 yields H( f, d) = 1 − ln λ, which is independent of d. This is the entropy of exponential distribution with parameter λ. For the opposite, assuming that H( f, d) is independent of d we have from Eq. 4 that ∂ H( f, d) = λ X (d)(ln λ X (d) + H( f, d) − 1). ∂d Setting the derivative equal to zero we take λ X (d) = e1−H( f,d) which is independent of d.   In other words, if the risk is constant, uncertainty with a deductible d does not depend on the deductible. This is the case of the exponential distribution. In the survival analysis and reliability theory literature, entropy H( f, d) has been presented as a measure of uncertainty of the residual lifetime distribution, i.e. given that an item has survived up to time t or reached “age” t. It is a dynamic version of the classical entropy (Ebrahimi 1996; Ebrahimi et al. 2007). In this context it is called residual lifetime entropy. In our treatment the deductible plays the role of “age”. Analogous measures of information for multivariate distributions when their supports are truncated progressively are proposed by Ebrahimi et al. (2007). From now on, H( f, d) will be called residual entropy with a deductible d or simply residual loss entropy.

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We now present some properties of the per-payment entropy H( f, d) of losses with a deductible d, extracted from the reliability and survival analysis literature. They are based on the mean residual loss given X > d ∞ F¯ X (x) dx μ(d) = E(X − d|X > d) = d F¯ X (d) and the risk function. The first two results, originally due to Ebrahimi (1996), in the framework of residual lifetimes, provide a further insight on the entropy of residual losses. We mainly focus on the interpretation of these properties in the context of actuarial science and loss distributions. 1. The entropy of residual losses with a deductible d is bounded by the mean residual loss. Specifically, it holds that H( f, d) ≤ 1 + ln μ(d),

2.

3. 4.

5.

6.

(5)

provided that μ(d) is finite (see Ebrahimi 1996). As stated in Ebrahimi (1996) this bound is derived from the maximum entropy principle for distributions over (0, ∞) with a given mean. See also Asadi et al. (2004). The per-payment residual loss entropy with a deductible d, H( f, d), is (i) increasing in d if λ X (x) is decreasing in x and (ii) decreasing in d if λ X (x) is increasing in x (see Ebrahimi 1996). If the mean residual loss is decreasing in d then the uncertainty H( f, d) is also decreasing in d. The opposite is not true (Ebrahimi et al. 2007). (i) If H( f, d) is increasing in d then its mean residual loss is also increasing. (ii) If the mean residual loss is decreasing then H( f, d) is decreasing in d (see Ebrahimi et al. 2007). Consider two insurance policies with risk functions λ1 (x) and λ2 (x) such that λ2 (x) = h(x)λ1 (x). Let also h(x) be increasing in x and 0 < h(x) < 1. Then if Policy 1 has a decreasing mean residual loss then Policy 2 has also a decreasing mean residual loss and consequently a decreasing H( f, d) (see Theorem 2.5 in Block et al. 1985). If X is absolutely continuous and H( f, d) is increasing in d then H( f, d) uniquely determines F X (x) (see Belzunce et al. 2004).

For the per-loss residual entropy we have the following bound in analogy with Result 1. Proposition 4 For the per-loss residual entropy it holds that Hl ( f, d) ≤ F¯ X (d)[1 + ln μ(d)] + H[F X (d), F¯ X (d)],

(6)

where μ(d) is the mean residual loss. Proof It follows from Eq. 3 and inequality in Eq. 5.

 

Example 3 Bound in Eq. 5 of H( f, d) for the exponential is 1 −  1 distribution   1 −τ τ τ ln λ while for the Weibull distribution is 1 + cd + ln c  τ , cd /τ , where

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∞  (a, x) = x ta−1 e−t dt (Abramowitz and Stegun 1972). For the Pareto distribution the bound is 1 + ln(λ + d) − ln(α − 1). Although we cannot algebrically compute the residual entropy for the Burr distribution we can find a bound ∞  (a)k (b )k zk of H( f, d) in terms of the hypergeometric function 2 F1 (a, b ; c; z) = , (c)k k! k=0

where(x)n = x · (x + 1) · . . . · (x + n − 1) (Abramowitz and Stegun 1972). This is   d−(τ α−1) 1 1 −τ 1 + ln τ α−1 2 F1 α, α − τ ; 1 + α − τ ; −λd . The derivation of the bound in Eq. 6 for Hl ( f, d) for the above mentioned distributions is straightforward.

4 The effect of a Policy Limit Suppose that losses X are not recorded for or above a certain policy (liability) limit u, u > 0. In this case the data W are truncated from above or, in other words, right truncated. Specifically, the random variable modeling the truncated from above losses is (Kluggman et al. 2008)  W(u) =

X, X 0. H(

(7)

2. If losses X have a decreasing reversed hazard function for all x, then past entropy ¯ f, u) increases for all u > 0. H( We have already mentioned that losses with truncation from above are different from losses with censoring from above (Kluggman et al. 2008), which are defined by the random variable  X, X < u equivalently V(u) = min{X, u}. V(u) = u, X ≥ u This is the per-loss case. Here, if the loss is X ≥ u, the insurance company pays an amount u. In other words the insurer pays a maximum amount of u on a claim. The random variable V(u) is not absolutely continuous. The following lemma gives the entropy expression of the random variable V(u). Lemma 6 The per-loss entropy of losses with a policy limit u or the per-loss past entropy is given by  u H¯ l ( f, u) = − f X (x) ln f X (x) dx − F¯ X (u) ln F¯ X (u) 0



= HX +

∞

f X (x) ln f X (x) dx − F¯ X (u) ln F¯ X (u).

u

Proof The result is obvious in view of definition in Eq. 1 and since the distribution of V(u) consists of a continuous part f X (x), x < u and a discrete part 1 − F X (u), at x = u.   Proposition 5 The following relations hold   (i) 3H X = Hl ( f, u) + H¯ l ( f, u) − H F X (d), F¯ X (d) ,   ¯ f, u) + H F X (d), F¯ X (d) , and (ii) H¯ l ( f, u) = F X (u) H(   (iii) Hl ( f, u) ≤ (ln u)F X (u) + H F X (u), F¯ X (u) . Proof Parts (i) and (ii) are easily shown by substitution. Part (iii) follows from (ii) and inequality in Eq. 7.   Example 5 The per-loss entropy of losses with a policy limit or the per-loss past entropy for the exponential distribution is H¯ l ( f, u) = (1 − ln λ)(1 − e−λu ), while for the Weibull distribution is  τ −1  τ τ H¯ l ( f, u) = (1 − ln c − ln τ )(1 − e−cu ) + γ +  (0, cuτ ) + ln c + e−cu τ ln u . τ

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The same entropy for the Pareto distribution is 1 H¯ l ( f, u) = 1 + + ln λ − ln α − α



λ λ+u

α   1 1 + − ln α + ln(λ + u) . α

Two other interesting quantities in loss distributions and actuarial mathematics are: W1 (u) = u − W, which describes the “remaining” of the coverage and W2 (u) = X − u, which describes the “remaining” of the claim. Because differential entropy is invariant under shifts (Cover and Thomas 1991) we have: ¯ f, u) and HW2 (u) = H X . HW1 (u) = H(

5 The effect of Deductible and Policy Limit Combination Let assume that d is the deductible and u the retention limit with d < u. In this case losses, i.e. payments to the policy holder are expressed (Kluggman et al. 2008) by ⎧ X≤d ⎨ 0, Y(d, u) = (X ∧ u) − (X ∧ d) = X − d, d < X ≤ u ⎩ u − d, X > u. Symbol ∧ denotes the minimum. The deductible d is applied after the implementation of the retention limit u, i.e. if the loss is greater than u then the maximum payment is u − d. The definition of Y(d, u) implies that we deal with the per-loss case. The random variable Y(d, u) is a mixed variable with an absolutely continuous part over the interval (0, u − d) and two discrete parts at 0 with probability mass F X (d) and at u − d with probability mass F¯ X (u). Lemma 7 The per-loss entropy of losses with a deductible d and a policy limit u, denoted by Hl ( f, d, u), is given by 

u

Hl ( f, d, u) = −F X (d) ln F X (d) −  = HX +

f X (x) ln f X (x) dx − F¯ X (u) ln F¯ X (u)

d d



f X (x) ln f X (x) dx +

0

∞

f X (x) ln f X (x) dx

u

−F X (d) ln F X (d) − F¯ X (u) ln F¯ X (u), d < u. Proof The proof is obvious since the “distribution” of Y(d, u) is given by ⎧ F X (d), ⎪ ⎪ ⎨ f X (y + d), fY(d,u) (y) = ¯ (u), F ⎪ ⎪ ⎩ X 0,

y=0 0< y u − d.  

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The following proposition relates Hl ( f, d, u) with the entropy under censoring from above H¯ l ( f, u) and the entropy under censoring from below Hl ( f, d). Proposition 6 The entropy Hl ( f, d, u) is related to the entropies H¯ l ( f, u) and Hl ( f, d) as follows Hl ( f, d, u) = H¯ l ( f, u) +



d

f X (x) ln f X (x) dx − F X (d) ln F X (d)

0

 = Hl ( f, d) +

∞

f X (x) ln f X (x) dx − F¯ X (u) ln F¯ X (u), d < u.

u

200

200

u 150

u 150 100

100 3

0.4 2 0.2 1 0.0 0

20 0

40 60

50

d

d

80 100

100

(a) exponential distribution with λ = 0.1

(b) Weibull distribution with and τ = 0.3

3 200 2 1 150

0

u

50 d 100

100

(c) Pareto distribution with λ = 1.2 and α = 0.3 Fig. 1 Hl ( f, d, u) for several distributions with d < u

c

= 1.3

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Example 6 The entropy under the previous combination scheme for the exponential distribution is Hl ( f, d, u) = e−λd (1 + λd − ln λ) − e−λu (1 − ln λ) − (1 − e−λd ) ln(1 − e−λd ), while for the Weibull distribution is  τ  τ  Hl ( f, d, u) = e−cd 1 − ln c − ln τ − (τ − 1) ln d + cdτ + ln 1 − e−cd τ

−e−cu (1 − ln c − ln τ − (τ − 1) ln u))   τ −1  τ  (0, cuτ ) −  (0, cdτ ) − ln 1 − e−cd , d < u. + τ For the Pareto distribution we have  α   1 λ Hl ( f, d, u) = 1 + − ln α − α ln λ + (α + 1) ln(λ + d) λ+d α  α   1 λ 1 + − ln α + ln(λ + u) − λ+u α α   α     λ λ ln 1 − , d < u. − 1− λ+d λ+d The corresponding graphs of Hl ( f, d, u) over the domain (0, 100) × (100, 200) ⊂ {(d, u) : 0 < d < u} are given in Fig. 1a–c, respectively. For the given values of the parameters, we see that as d and u increase, Hl ( f, d, u) tends to be constant. It is easy to see, from Lemma 7, that limd→0,u→∞ Hl ( f, d, u) = H X . This is also verified for the distributions treated in Example 6.

6 Residual and Past Entropies for Survival Models In this section we derive residual and past entropy expressions for several survival models such as the proportional and reversed hazards models and the frailty model. Let X and Y be random variables with distribution functions F and G, respectively. Let also f and g the pdf’s of X and Y, respectively. Proportional Hazards Model Let X and Y satisfy the proportional hazards model θ  ¯ ¯ for all x > 0 and θ > 0 or λG (x) = θλ F (x). G(x) = F(x)

(8)

In other words, the hazards of Y are proportional to the hazards of X. See Cox (1972) for details. Theorem 1 Under the proportional hazard model given in Eq. 8 the residual entropy H(g, d) of the random variable Y is given by  H(g, d) = 1 − 1/θ − ln θ −



1

ln 0

f (x) ¯ F(d)

¯ ¯ ¯ obtained from y = F(x)/ F(d). where x = F −1 (1 − y F(d)),



dyθ ,

(9)

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Methodol Comput Appl Probab (2014) 16:79–99

Proof We have 

∞

g(x) g(x) ln dx ¯ ¯ G(d) G(d) d  ∞ θ −1 θ −1 ¯ ¯ θ( F(x)) f (x) θ( F(x)) f (x) ln dx =− θ θ ¯ ¯ ( F(d)) ( F(d)) d ⎡ ⎤  θ −1 θ −1  ∞  ¯ ¯ F(x) F(x) f (x) ⎣ f (x) ⎦ =− dx ln θ θ ¯ ¯ ¯ ¯ F(d) F(d) F(d) F(d) d

H(g, d) = −

¯ ¯ which by a change of variable y = F(x)/ F(d) takes the desired form.

 

Note that Eq. 9 of H(g, d) involves d. The actual dependence of H(g, d) on d relies on model F. This is in contrast with the Kullback–Leibler residual discrimination KL measure I X,Y (d) between X and Y which does not depend on d as it is deducted from Theorem 8 of Vonta and Karagrigoriou (2010). Under the present general model, uncertainty for lifetimes of losses beyond d depends, in general, on d. A similar result can be obtained if we consider, in the same framework, the past ¯ entropy H(g, u) of Y. We omit details. Example 7 Let X follow an exponential distribution with parameter λ. It is easy to see that under the proportional hazard model given in Eq. 8 the residual entropy of Y, H(g, d), is given by H(g, d) = 1 − ln(λθ). This is independent of d. Let X follow a Weibull distribution with parameters c andτ . The residual entropy  τ −1 τ H(g, d) is given by H(g, d) = 1 − 1/θ + ecθ d Ei(−cθ dτ ) − ln c τ θ dτ − cθ dτ + ln τ . Let X follow a Pareto distribution with parameters α and λ. The residual entropy H(g, d) is given by H(g, d) = 1 + 1/αθ − ln(αθ) + ln(λ + x). Proportional Reversed Hazards Model Let X and Y be random variables with distribution functions F and G respectively, which satisfy the proportional reversed hazards model G(x) = (F(x))θ for all x > 0 and θ > 0.

(10)

See Di Crescenzo (2000), Gupta and Gupta (2007) and Sankaran and Gleeja (2008) for details. Theorem 2 Under the proportional reversed hazards model given in Eq. 10 the past ¯ entropy H(g, u) of the random variable Y is given by ¯ H(g, u) = 1 − 1/θ − ln θ −





1

ln 0

where x = F −1 (yF(u)), obtained from y = F(x)/F(u).

f (x) F(u)



dyθ ,

(11)

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95

Proof We have ¯ H(g, u) = −



u

0



g(x) g(x) ln dx G(u) G(u)

θ(F(x))θ −1 f (x) θ(F(x))θ −1 f (x) ln dx (F(u))θ (F(u))θ 0       u  F(x) θ −1 f (x) F(x) θ −1 f (x) ln θ dx θ =− F(u) F(u) F(u) F(u) 0 =−

u

which by a change of variable y = F(x)/F(u) takes the desired form.

 

¯ ¯ Again, Eq. 11 of H(g, u) involves u. The actual dependence of H(g, u) on u relies on model F. This is in contrast with the Kullback–Leibler residual discrimination KL measure I X,Y (u) between X and Y which does not depend on u as it is deducted from Theorem 9 of Vonta and Karagrigoriou (2010). Striking is the similarity of Eqs. 9 and 11. A similar result can be obtained if we consider, in the same framework, the residual entropy H(g, d) of Y. Again we omit the details. Example 8 Let X follow an exponential distribution with parameter λ. It is easy to ¯ see that under the reversed hazard model given in Eq. 10 the past entropy H(g, u) is given by 1 e−λu (eλu − 1)2 F1 (1, θ + 1; θ + 2; 1 − e−λu ) eλu − 1 ¯ H(g, u) = 1 − − + ln . θ θ +1 θ Let X follow a Pareto distribution with parameters λ and a. Its past entropy is given by   α  1 λ H(g, u) = 1 − − ln θ − ln α − (α + 1) ln λ + ln 1 − θ λ+u   a  a    1 λ λ 1− F 1, θ + 1; θ + 2; 1 − + 2 1 α(θ + 1) λ+u λ+u a  λ . − ln λ+u Frailty Model Let Y follow a frailty model under which ¯ G(y) = e−Q(θ (y)) , θ > 0.

(12)

where Q(·) is a concave and increasing function with Q(0) = 0 and Q(∞) = ∞, (x) is the baseline cumulative hazard function of X with (∞) = ∞. See Sankaran and Gleeja (2008), Vonta (1996) and Vonta and Karagrigoriou (2010) for details. Next we provide expressions for the residual and past entropies of Y, H(g, d), in terms of Q and the cumulative hazard of X. We also specialize the expressions for common frailty models.

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Lemma 8 The residual and past entropies of Y under the frailty model given in Eq. 12 are given by    G(d) ¯ dz 1 ¯ dz (13) ln − H(g, d) = ln G(d) − ¯ dy G(d) 0 and

   G(u) ¯ ¯ dz 1 G(u) ¯ H(g, u) = − ln − ln G(u) + dz, G(u) G(u) 0 dy

(14)

−Q(θ (u)) ¯ ¯ , G(u) , and dz/ dy = respectively, where G(d) = e−Q(θ (d))  = e ¯ −zQ (θ (y))θ λ(y) with y = −1 Q−1 (− ln z)/θ , obtained from z = G(y) = e−Q(θ (y)) .

¯ Proof We shall illustrate the proof for H(g, d). The steps for H(g, u) are similar. By definition we have  ∞ g(y) g(y) ln dy H(g, d) = − ¯ ¯ G(d) G(d) d  ∞ −Q(θ (y))  e Q (θ (y))θλ(y) e−Q(θ (y)) Q (θ (y))θλ(y) =− ln dy. ¯ ¯ G(d) G(d) d ¯ By a change of variable z = e−Q(θ (y)) = G(y) and since dz/ dy = −zQ (θ (y))θλ(y) and lim y→∞ e−Q(θ (y)) = 0 we obtain     G(d) ¯ 1 dz ¯ H(g, d) = − − ln G(d) dy − ¯ dy G(d) d    G(d) ¯ 1 dz ¯ = ln G(d) − ln − dz. ¯ dy G(d) 0   θ  ¯ ¯ . Example 9 Let Y satisfy the Cox model, i.e. it holds Q(x) = x or G(x) = F(x) Then the frailty residual entropy H(g, d) given by Eq. 13 yields, after some algebra, the proportional hazard residual entropy Eq. 9. The same is true for frailty past ¯ entropy H(g, u) and the proportional reversed hazard past entropy Eq. 11. Example 10 Let Q(x, c) = ln(1 + cx)/c for c > 0 which corresponds to a Gamma distributed frailty with mean 1 and variance c (cf. Vonta and Karagrigoriou 2010). Let us take the variance c to be equal to one and the baseline hazard function to be equal to λ > 0, i.e. X is exponential with parameter λ and (x) = λx. Then ¯ G(y) = e− ln(1+θλy) = 1/(1 + θλy), z = 1/(1 + θλy) and by Eqs. 13 and 14 we obtain H(g, d) = 2 + ln(1 + θλd) − ln(θλ) and 1 [ln(θλ) − ln(θλu) − ln(1 + θλu) − 2] . θλu ¯ Note that H(g, d) ≥ 0 for all θ, λ, d > 0, while H(g, u) ≤ 0 for u > 1 and all θ, λ > 0. ¯ H(g, u) =

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97

7 Application—Numerical Illustration In this section we consider an application of the concepts, introduced above, of entropies with a deductible and/or a policy limit using a well known and classical data set in insurance science, namely that of the Danish fire insurance losses from the year 1980–1990 (cf. McNeil 1997; Resnick 1997; Pigeon and Denuit 2011). Losses are ranged from 1.0 to 263.250 MDKK (millions of Danish Krone). The average loss is 3.385 MDKK while the 25 % of losses are smaller than 1.321 MDKK and the 75 % are smaller than 2.967 MDKK. These entropies are also known as residual and past entropies. As stated before, the entropy of a distribution can be used as a measure of uncertainty or variability. As such we fit the Weibull distribution to the data set and study the uncertainty in terms of deductibles and policy limits. By fitting the Weibull distribution to the data we obtained cˆ = 0.3192 and τˆ = 0.9585 as the maximum likelihood estimators of the shape and scale parameters of the distribution. Table 1 presents the values of the initial entropy H X , the perpayment and per-loss entropies with a deductible d, H( f, d) and Hl ( f, d) respec¯ f, u) and tively, the per-payment and per-loss entropies with a policy limit u, H( ¯ Hl ( f, u) respectively, and the per-loss entropy with a deductible d and a policy limit u Hl ( f, d, u), for several values of d and u. We observe that as d increases H( f, d) ¯ f, u) increases while Hl ( f, d) decreases. As u increases H¯ l ( f, u) increases while H( decreases. Hl ( f, d, u) increases as d and u increase. Per-loss entropies Hl ( f, d) and Hl ( f, u) are the real losses for the insurance company in the case we have a deductible d and a policy limit u, respectively. The above mentioned results reveal that as d increases, the uncertainty of losses for the insurance company decreases as the company does not pay for smaller claims believed to be abundant. As u increases

Table 1 Entropy values for Weibull distribution with cˆ = 0.3192 and τˆ = 0.9585 d

u

HX

H( f, d)

Hl ( f, d)

¯ f, u) H(

H¯ l ( f, u)

Hl ( f, d, u)

1.1

10.0 15.0 20.0 25.0 10.0 15.0 20.0 25.0 10.0 15.0 20.0 25.0 10.0 15.0 20.0 25.0 10.0 15.0 20.0 25.0

2.20865

2.23697

1.09104

2.20865

2.23842

1.06212

2.20865

2.23981

1.03212

2.20865

2.24113

1.00127

2.20865

2.24240

0.96975

2.31601 2.25345 2.22484 2.21409 2.31601 2.25345 2.22484 2.21409 2.31601 2.25345 2.22484 2.21409 2.31601 2.25345 2.22484 2.21409 2.31601 2.25345 2.22484 2.21409

2.08275 2.17672 2.20041 2.20649 2.08275 2.17672 2.20041 2.20649 2.08275 2.17672 2.20041 2.20649 2.08275 2.17672 2.20041 2.20649 2.08275 2.17672 2.20041 2.20649

2.07928 2.17326 2.19694 2.20302 2.09548 2.18946 2.21314 2.21923 2.11294 2.20691 2.23059 2.23668 2.13136 2.22534 2.24902 2.25511 2.15055 2.24452 2.26821 2.27429

1.2

1.3

1.4

1.5

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the uncertainty for the insurance company increases as it has to pay greater amounts. These results are consistent with the theoretical results of Sections 4 and 5.

8 Conclusions The role of this paper is to investigate the effect that several partial insurance schemes, such as inflation, truncation and censoring from above, truncation and censoring from below, have on Shannon’s entropy. These situations frequently appear in actuarial practice as a consequence of the use of deductibles and policy limits. The use of a deductible leads to truncation and censoring from below while policy limit lead to truncation and censoring from above. It is natural to consider entropies of losses with a deductible and a policy limit respectively. We proved that inflation increases the entropy, which means that under inflation we have more uncertainty compared to the initial state. We differentiated between per-payment and per-loss entropies of losses. We also identified the entropies of losses with a deductible and a policy limit with the residual and past entropies of lifetimes. Residual entropy of losses with a deductible is not always smaller than the initial entropy as one might expect. If the initial entropy is negative then the use of a deductible results in a per-payment residual entropy which is always less than the per-loss. The residual entropy of losses with a deductible d is (i) independent of the deductible if and only if the risk function of losses is constant, and (ii) increases (decreases) in d if the risk function of losses decreases (increases). The entropy of losses with a policy limit u increases in u if the risk function decreases. Several relationships between the entropies under these schemes were presented. The numerical illustration showed that d affects H( f, d), Hl ( f, d) and Hl ( f, d, u). The policy limit u has a minor effect on H¯ l ( f, u). We also computed the residual and past entropy of the proportional hazards model, the proportional reversed hazards model and the frailty model. These are models used when the study population can not be assumed to be homogeneous but must be considered as an heterogeneous sample. Here again the entropy, in general, depends on the deductible and the policy limit.

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